Abstract Within the rough path framework, we prove the continuity of the solution to random differential equations driven by a one dimensional fractional Brownian motion with respect to the Hurst parameter H when H ∈ ( 1 / 3 , 1 / 2 ] .

Abstract The global well-posedness as well as long-term behavior in terms of mean random attractors and invariant measures are investigated for a class of stochastic discrete reaction-diffusion equations defined on Z k with a family of superlinear noise. The existence and uniqueness of weak pullback mean random attractors for the mean random dynamical system associated with the non-autonomous equations

Abstract The article is devoted to the existence and Hyers-Ulam stability of the almost periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal condition. The investigation is mainly based on the semigroups of operators method and Mönch fixed points method, as well as the basic theory of Hyers-Ulam stability.

Abstract We study the asymptotic behavior of a real-valued diffusion whose nonregular drift is given as a sum of a dissipative term and a bounded measurable one. We prove that two trajectories of that diffusion converge almost sure to one another at an exponential explicit rate as soon as the dissipative coefficient is large enough. A similar result in Lp is obtained.

In this paper, we discuss a class of stochastic nonlinear partial differential equation of Burgers type driven by pseudo differential operator ( Δ + Δ α ) for α ∈ ( 0 , 2 ) , and fractional Brownian sheet. The existence and uniqueness of an Lp-valued (local) solution is established for the initial valued problem to the equation.

We investigate the joint distribution and the multivariate survival functions for the maxima of an Ornstein-Uhlenbeck (OU) process in consecutive time-intervals. A PDE method, alongside an eigenfunction expansion is adopted, with which we first calculate the distribution and the survival functions for the maximum of a homogeneous OU-process in a single interval. By a deterministic time-change and a

The aim of this manuscript is to analyze the existence of mild solution of non-instantaneous impulsive Hilfer fractional stochastic differential equations (NIHFSDEs) driven by fractional Brownian motion (fBm). Sufficient conditions for a class of NIHFSDEs of order 0 < β < 1 and of type 0 ≤ α ≤ 1 driven by fBm is derived with the help of fractional calculus, stochastic theory, fixed point theorem and

In the paper, we consider some piecewise deterministic Markov process, whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two dynamical systems have been already investigated in our recent papers. We now aim

We introduce and study a marked Poisson random measure on a σ-compact Hausdorff space. The underlying parameters of this measure are changing in accordance with the evolution of some stochastic process. This random measure (also known as a modulated measure) resembles those of the conventional Poisson random measure. Among many applications, the one about Poisson random measures modulated by semi-Markov

Let W denote the Brownian motion. For any exponentially bounded Borel function g the function u defined by u ( t , x ) = E [ g ( x + σ W T − t ) ] is the stochastic solution of the backward heat equation with terminal condition g. Let u n ( t , x ) denote the corresponding approximation generated by a simple symmetric random walk with time steps 2 T / n and space steps ± σ T / n where σ > 0 . For a

Kolmogorov’s converse inequality is proved for dependent random variables in order to obtain necessary and sufficient conditions for weak and strong laws (for not necessarily strictly stationary sequences) under strong dependence without rate assumptions.

Large deviation principle by the weak convergence approach is established for the stochastic nonlinear Schrödinger equation in one-dimension and as an application the exit problem is investigated.

We consider a new class of one-dimensional diffusion processes with self-similar random potentials. The self-similar random potential has different exponents to the left and the right hand sides of the origin. We show that, because of the difference between the two exponents, the long-time behaviors of our process on the left and the right hand sides of the origin are quite different from each other

On the global platform of emerging infectious diseases, dengue fever added a serious health concern, especially in the tropical and subtropical countries with poor health services. Due to unavailability of proper vaccination, primary prevention of dengue is possible at social and personal levels by controlling the propagation of vectors as well as avoiding the bites of infected ones, respectively.

Abstract We formulate ill-posedness of inverse problems of estimation and prediction of Coronavirus Disease 2019 (COVID-19) outbreaks from statistical and mathematical perspectives. This is by nature a stochastic problem, since e.g., random perturbation in parameters can cause instability of estimation and prediction. This leaves us with a plenty of possible statistical regularizations, thus generating

This paper is concerned with existence, uniqueness, and approximate controllability of mild solutions to second-order non-autonomous stochastic impulsive differential systems. The impulsive differential systems may provide mathematical models to the phenomena having discontinuous jumps. The main results are carried out by using the generalized Banach contraction principle, evolution family, and stochastic

We study optimal control for mean-field forward–backward stochastic differential equations with payoff functionals of mean-field type. Sufficient and necessary optimality conditions in terms of a stochastic maximum principle are derived. As an illustration, we solve an optimal portfolio with mean-field risk minimization problem.

We prove an anticipative sufficient stochastic minimum principle in a jump process setup with initially enlarged filtrations. We apply the result to several portfolio selection problems like mean and minimal variance hedging under enlarged filtrations. We also investigate utility maximizing portfolio selection under future information. Contrarily to classical optimization methods like dynamic programing

In this article, we derive the risk-neutral measures of the stock options price and volatility by incorporating a simple constrained minimization of the Kullback measure of relative information. We obtain a second-order parabolic partial differential equation, the generalized Black–Scholes equation based on the theoretical analysis when the underlying financial asset is estimated using a stochastic

In this paper, stochastic integral equations characterized by fractional Brownian motion have been studied. The fractional stochastic integral equation has been solved by second kind Chebyshev wavelets. The convergence and error analysis have been discussed for the efficiency of the discussed method. In addition, two illustrative examples have been solved to examine the efficiency and accuracy of the

In this paper, by using Girsanov’s transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index H ∈ ( 1 / 2 , 1 ) .

We study nonzero-sum stochastic differential games with risk-sensitive discounted cost criteria. Under fairly general conditions on drift term and diffusion coefficients, we establish a Nash equilibrium in Markov strategies for the discounted cost criterion. We achieve our results by studying relevant systems of coupled HJB equations.

This paper considers an optimal time-consistent proportional reinsurance problem with constraints on the strategies under the mean-variance criterion for an insurer. It is assumed that the surplus process is described by a compound Poisson risk model with dependent classes of insurance business named thinning-dependence structure, in which the stochastic sources related to claim occurrence are classified

In this correspondence, for an array of rowwise independent random elements { V n , k , 1 ≤ k ≤ k n , n ≥ 1 , k n → ∞ } taking values in a real separable Rademacher type p   ( 1 ≤ p ≤ 2 ) Banach space and a sequence of positive constants { c n , n ≥ 1 } , the main result provides conditions for the complete convergence result ∑ n = 1 ∞ c n P ( max 1 ≤ k ≤ k n | | ∑ i = 1 k V n , i | | > ε ) < ∞   for

The objective of this paper is to investigate the existence of mild solutions and optimal controls for a class of fractional neutral stochastic differential equations (NSDEs) driven by fractional Brownian motion and Poisson jumps in Hilbert spaces. First, we establish a new set of sufficient conditions for the existence of mild solutions of the aforementioned fractional systems by using the successive

This paper explores the relationship between non-Markovian fully coupled forward–backward stochastic systems and path-dependent PDEs. The definition of classical solution for the path-dependent PDE is given within the framework of functional Itô calculus. Under mild hypotheses, we prove that the forward–backward stochastic system provides the unique classical solution to the path-dependent PDE.

The present paper addresses the issue of choosing an optimal dynamic reinsurance policy, which is state-dependent, for an insurance company that operates under multiple insurance business lines. For each line, the Cramer–Landberg model is adopted for the risk process and one of the contracts such as Proportional reinsurance, excess-of-loss reinsurance (XL) and limited XL reinsurance (LXL) is intended

Multivariate Bessel processes (Xt,k)t≥0 describe interacting particle systems of Calogero-Moser-Sutherland type and are related with β-Hermite and β-Laguerre ensembles. They depend on a root system and a multiplicity k. Recently, several limit theorems were derived for k→∞ with fixed starting point. Moreover, the SDEs of (Xt,k)t≥0 were used to derive strong laws of large numbers for k→∞ with starting

After establishing the moderate deviation principle by the classical Azencott method, we prove the Strassen’s compact law of the iterated logarithm (LIL) for a class of stochastic partial differential equations (SPDEs). As an application, we obtain this type of LIL for two population models known as super-Brownian motion and Fleming-Viot process. In addition, the classical LIL is shown for the class

We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1/2

In this article, based on the tool of coupling method, Girsanov’s theorem and a stopping argument, we establish the exponential ergodicity for white noise driven parabolic stochastic partial differential equations when the coefficients are non-Lipschitz continuous.

The evolution of biofilms on surfaces of medical implants, food, machinery, etc. may be modeled via reaction-diffusion partial differential equations. According to Allee effect, the microbes growth rate is positively correlated with the microbes density. Thus, instead of using Fisher’s equation, we employ the bistable Allen–Cahn equation for the modeling of biofilm formation. We work in space dimension

Many engineering materials of interest are polycrystals: an aggregate of many crystals with size usually below 100 μm. Those small crystals are called the grains of the polycrystal, and are often equiaxed. However, because of processing, the grain shape may become anisotropic; for instance, during recrystallization or phase transformations, the new grains may grow in the form of ellipsoids. Heavily

The goal of this paper is to establish the Lipschitz and W2,∞ estimates for a second-order parabolic PDE ∂tu(t,x)=12Δu(t,x)+f(t,x) on Rd with zero initial data and f satisfying a Ladyzhenskaya–Prodi–Serrin type condition. Following the theoretic result, we then give two applications. The first is to discuss the regularity of the stochastic heat equations, and the second is to discuss the Sobolev differentiability

We study an inverse problem for the first-passage place of a one-dimensional diffusion process X(t) (also with jumps), starting from a random position η∈[a,b]. Let be τa,b the first time at which X(t) exits the interval (a,b), and πa=P(X(τa,b)≤a) the probability of exit from the left of (a,b). Given a probability q∈(0,1), the problem consists in finding the density g of η (if it exists) such that πa=q

We propose a nonparametric estimation of random effects from the following small fractional diffusions dXi(t)=βib(Xi(t))dt+εdWi,H(t), Xi(0)=x0i, 0≤t≤T. For i=1,⋯,N, βi is random variable and (Wi,H(t), t≥0, i=1,⋯,N) is are standard fractional Brownian motions with a common known Hurst index H∈(1/2,1). The asymptotic behavior of the proposed estimators is established when ε and N are, respectively, sufficiently

We study the locally uniform convergence from a family of pullback random attractors to a deterministic attractor. We establish criteria by using joint convergence of the cocycles, collective local compactness and deterministic recurrence of the random attractors. All three conditions are verified in the content of stochastic non-autonomous Magneto-hydrodynamics equations under the weak assumption

The Adomian decomposition method (ADM) is a powerful tool to solve several nonlinear functional equations and a large class of initial/boundary value problems. In this paper, we discuss a probabilistic approach to compute the Adomian polynomials (AP’s), which is the main part of the ADM. We provide a probabilistic interpretation for the AP’s, both for the one-variable and the multivariable case. We

This paper deals with the limiting behavior of non-autonomous stochastic reaction-diffusion equations without uniqueness on unbounded narrow domains. We prove the existence and upper semicontinuity of random attractors for the equations on a family of unbounded (n + 1)-dimensional narrow domains, which collapses onto an n-dimensional domain. Since the solutions are non-uniqueness, which leads to a

We provide conditions implying finite-time blowup of positive weak solutions to the semilinear equation du(t,x)=[Δαu(t,x)+Ku(t,x)+u1+β(t,x)]dt+μu(t,x) dBtH,u(0,x)=f(x),x∈R d,t≥0, where α∈(0,2],K∈R,β>0,μ≥0 and H∈[12,1) are constants, Δα is the fractional power −(−Δ)α/2 of the Laplacian, (BtH) is a fractional Brownian motion with Hurst parameter H, and f≥0 is a bounded measurable function. To achieve

In this paper, we consider a stochastic tuberculosis model with two kinds of treatments, that is, treatment at home and treatment in hospital. Firstly, we obtain sufficient conditions for extinction and persistence in the mean of the diseases, then in the case of persistence, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions

Complementary regularity between the integrand and integrator is a well known condition for the integral ∫0Tf(r) dg(r) to exist in the Riemann-Stieltjes sense. This condition also applies to the multi-dimensional case, in particular the 2 D integral ∫[0,T]2f(s,t) dg(s,t). In the paper, we give a new condition for the existence of the integral under the assumption that the integrator g is a Volterra

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the results available

In the present paper we consider a family of non-Volterra quadratic stochastic operators depending on a parameter α and study their trajectory behaviors. We find all fixed and periodic points for a non-Volterra quadratic stochastic operator on a finite-dimensional simplex. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.

The aim of this article is to consider the existence, uniqueness and uniformly exponential p-stability of the solution for ∇-delay stochastic dynamic equations on time scales via Lyapunov functions. This work can be considered as a unification and generalization of stochastic difference and stochastic differential time-varying delay equations.

In this work, we study the full randomized versions of Airy, Hermite and Laguerre differential equations, which depend on a random variable appearing as an equation coefficient as well as two random initial conditions. In previous contributions, the mean square stochastic solutions to the aforementioned random differential equations were constructed via the Fröbenius method, under the assumption of

In this article we define and investigate statistical operators and an entropy functional for Bernstein stochastic processes associated with hierarchies of forward-backward systems of decoupled deterministic linear parabolic partial differential equations. The systems under consideration are defined on open bounded domains D⊂Rd of Euclidean space where d∈N+ is arbitrary, and are subject to Neumann

In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is the multi-factor Heston stochastic volatility model. We obtain pricing formulas for the weighted variance swap and approximate expressions for the weighted volatility swap. The bounds of the arbitrage-free

We are interested in the optimal filter in a continuous time setting. We want to show that the optimal filter is stable with respect to its initial condition. We reduce the problem to a discrete time setting and apply truncation techniques. Due to the continuous time setting, we need a new technique to solve the problem. In the end, we show that the forgetting rate is at least a power of the time t

In this paper, we study the dynamical behavior of a multigroup SIQS epidemic model, which is formulated as a piecewise deterministic Markov process. Sufficient conditions for extinction and persistence in the mean of the diseases are obtained. In addition, in the case of diseases persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by

In this article, we study the following stochastic heat equation ∂tu(t,x)=Lu(t,x)+Ḃ, u(0,x)=0, 0≤t≤T, x∈Rd, where L is the generator of a Lévy process X in Rd, B is a fractional-colored Gaussian noise with Hurst index H∈(12, 1) in the time variable and spatial covariance function f which is the Fourier transform of a tempered measure μ. After establishing the existence of solution for the stochastic

In this paper, Wang’s Harnack inequalities and super Poincaré inequality for generalized Cox-Ingersoll-Ross model are obtained. Since the noise is degenerate, the intrinsic metric has been introduced to construct the coupling by change of measure. By using isoperimetric constant, some optimal estimate of the rate function in the super Poincaré inequality for the associated Dirichlet form is also obtained

Abstract The goal of this paper is to study a stochastic game connected to a system of forward-backward stochastic differential equations (FBSDEs) involving delay and noisy memory. We derive sufficient and necessary maximum principles for a set of controls for the players to be a Nash equilibrium in the game. Furthermore, we study a corresponding FBSDE involving Malliavin derivatives. This kind of

Abstract In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular

The main result of the paper deals with weak compactness in distribution of weak solutions sets to stochastic differential inclusions defined by set-valued stochastic integrals presented in the paper [Kisielewicz, M., Michta, M. (2017 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449(2):1892–1910. DOI: 10.1016/j.jmaa.2017.01.013.[Crossref]

Let ξ:Ω×Rn→R be zero mean, mean-square continuous, stationary, Gaussian random field with covariance function r(x)=E[ξ(0)ξ(x)] and let G:R→R such that G is square integrable with respect to the standard Gaussian measure and is of Hermite rank d. The Breuer-Major theorem in it’s continuous setting gives that, if r∈Ld(Rn), then the finite dimensional distributions of Zs(t)=1(2s)n/2∫[−st1/n,st1/n]n[G

This paper provides a theoretical analysis on the impacts of using a suboptimal information set for the estimation of the pricing kernel and, more in general, for the validity of the fundamental theorems of asset pricing. While inferring the risk-neutral measure from options data provides a naturally forward-looking estimate, extracting the real world measure from historical returns is only partially

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some

We study Brownian motion in the setting of quaternion analysis. We give a quaternion version of the Itô’s integral.

In this work, we develop polynomial processes where the coefficients of the process may depend on time. A full characterization of this model class is given by means of their semimartingale characteristics. While in the time-homogeneous case, moments can be calculated easily through matrix exponentials, we show that in the inhomogeneous case this is no longer possible. As an alternative we explore